The Basic Theory of Ordering Relations

What follows is the briefest possible summary of the order-theoretic
notions used in the main text. For a good introduction to this
material, see Davey & Priestley [1990]. More advanced treatments
can be found in Gratzer [1998] and Birkhoff [1967].

A partial ordering—henceforth, just an ordering—on a
set P is a reflexive, anti-symmetric, and transitive binary
relation ⊴ on
P. Thus, for all p, q, r ∈
P, we have

p ⊴ p

p ⊴ q and q ⊴ p only if
p = q.

if p ⊴ q and q ⊴ r then
p ⊴ r

If p ⊴ q, we speak of p as being less
than, or belowq, and of q as being
greater than, or abovep, in the
ordering.

A partially ordered set, or poset, is a pair
(P, ⊴ )
where P is a set and
⊴
is a specified ordering on P. It is usual
to let P denote both the set and the structure, leaving
⊴
tacit wherever
possible. Any collection of subsets of some fixed set X,
ordered by set-inclusion, is a poset; in particular, the full power
set
℘(X) is
a poset under set inclusion.

Let P be a poset. The meet, or greatest lower
bound, of p, q ∈ P, denoted by
p∧q, is the greatest element of P—if there is one—lying below both p and q. The
join, or least upper bound, of p and
q, denoted by p∨q, is the least element
of P—if there is one—lying above both p
and q. Thus, for any
elements p, q, r of P, we have

Note that if the set P =
℘(X),
ordered by set-inclusion, then p∧q =
p∩q and p∨q =
p∪q. However, if P is an arbitrary
collection of subsets of X ordered by inclusion, this need not
be true. For instance, consider the collection P of all
subsets of X = {1,2,...,n} having even cardinality.
Then, for instance,
{1,2}∨{2,3} does not exist in P, since there is no
smallest set of 4 elements of X containing {1,2,3}.
For a different sort of example, let X be a vector space and
let P be the set of subspaces of X. For
subspaces M and N, we have

M∧N =
M∩N, but M∨N =
span(M∪N).

The concepts of meet and join extend to infinite subsets of a poset
P. Thus, if A⊆P, the meet of A
is the largest element (if any) below A, while the join of
A is the least element (if any) above A. We denote
the meet of A by
∧A or by
∧a∈Aa.
Similarly, the join of A is denoted by
∨A or by
∨a∈Aa.

A lattice is a poset (L, ⊴ ) in which every
pair of elements has both a meet and a join. A complete
lattice is one in which every subset of L has a
meet and a join. Note that
℘(X) is a complete lattice with respect to set
inclusion, as is the set of all subspaces of a vector space. The set of
finite subsets of an infinite set X is a lattice, but not a
complete lattice. The set of subsets of a finite set having an even
number of elements is an example of a poset that is not a lattice.

The power set lattice
℘(X), for instance, is distributive (as is any
lattice of sets in which meet and join are given by set-theoretic
intersection and union). On the other hand, the lattice of subspaces of
a vector space is not distributive, for reasons that will become clear
in a moment.

A lattice L is said to be bounded iff it contains a
smallest element 0 and a largest element 1. Note that any complete
lattice is automatically bounded. For the balance of this appendix,
all lattices are assumed to be bounded, absent any indication
to the contrary.

A complement for an element p of a (bounded)
lattice L is another element q such that p
∧ q =
0 and p
∨
q = 1.

In the lattice
℘(X), every element has exactly one
complement, namely, its usual set-theoretic complement. On the other
hand, in the lattice of subspaces of a vector space, an element will
typically have infinitely many complements. For instance, if L
is the lattice of subspaces of 3-dimensional Euclidean space, then a
complement for a given plane through the origin is provided by any line
through the origin not lying in that plane.

Proposition:
If L is distributive, an element of L can have at
most one complement.

Proof:
Suppose that q and r both serve as complements for
p. Then, since L is distributive, we have

q

=

q∧1

=

q
∧
(p∨r)

=

(q∧p)
∨
(q∧r)

=

0
∨
(q∧r)

=

q∧r

Hence, q ⊴ r. Symmetrically, we have r ⊴ q; thus,
q = r.

Thus, no lattice in which elements have multiple complements is
distributive. In particular, the subspace lattice of a vector space (of
dimension greater than 1) is not distributive.

If a lattice is distributive, it may be that some of its
elements have a complement, while others lack a complement. A
distributive lattice in which every element has a complement is called
a Boolean lattice or a Boolean algebra. The basic
example, of course, is the power set
℘(X) of a
set X. More generally, any collection of subsets of X
closed under unions, intersections and complements is a Boolean
algebra; a theorem of Stone and Birkhoff tells us that, up to
isomorphism, every Boolean algebra arises in this way.

In some non-uniquely complemented (hence, non-distributive)
lattices, it is possible to pick out, for each element p, a
preferred complement p′ in such a way that

if p ⊴ q then q′
⊴
p′

p′′ = p

When these conditions are satisfied, one calls the mapping
p→p′ an orthocomplementation on
L, and the structure (L, ⊴ ,′) an
orthocomplemented lattice, or an ortholattice for
short.

Note again that if a distributive lattice can be orthocomplemented
at all, it is a Boolean algebra, and hence can be orthocomplemented in
only one way. In the case of L(H) the
orthocomplementation one has in mind is M →
M⊥ where
M⊥ is defined as in Section 1 of the
main text. More generally, if V is any inner product
space (complete or not), let L(V) denote the
set of subspaces M of V such that
M = M⊥⊥ (such a
subspace is said to be algebraically closed). This again is a complete
lattice, orthocomplemented by the mapping M →
M⊥.

There is a striking order-theoretic characterization of the lattice
of closed subspaces of a Hilbert space among lattices
L(V) of closed subspaces of more general
inner product spaces. An ortholattice L is said to be
orthomodular iff, for any pair p, q in L
with p ⊴ q,

(OMI) (q∧p′)∨p =
q.

Note that this is a weakening of the distributive law. Hence, a
Boolean lattice is orthomodular. It is not difficult to show that if
H is a Hilbert space, then
L(H) is orthomodular. The striking converse
of this fact is due to Amemiya and Araki [1965]:

Theorem:
Let V be an inner product space (over
R, C or the quaternions) such that
L(V) is orthomodular. Then V
is complete, i.e., a Hilbert space.

Let P and Q be posets. A mapping
f : P → Q is order
preserving iff for all p,q ∈ P, if
p ⊴ q then f(p) ⊴ f(q).

A closureoperator on a poset P is an
order-preserving map
cl : P → P such
that for all p ∈ P,

cl(cl(p)) =
cl(p)

p
⊴
cl(p).

Dually, an interior operator on P is an
order-preserving mapping
int : P → P on
P such that for all p ∈ P,

int(int(p)) =
int(p)

int(p)
⊴
p

Elements in the range of cl are said to be
closed; those in the range of int are said to
be open. If P is a (complete) lattice, then the set
of closed, respectively open, subsets of P under a closure or
interior mapping is again a (complete) lattice.

By way of illustration, suppose that
O
and
C
are collections of
subsets of a set X with
O
closed under arbitrary unions and
C
under arbitrary
intersections. For any set A ⊆ X, let

cl(A) =
∩{C∈C |
A ⊆ C}, and

int(A) = ∪{O∈O |
O ⊆
A}

Then cl and int are interior
operators on
℘(X), for which the closed and open sets are
precisely
C
and
O, respectively.
The most familiar example, of course, is that in which
O,
C are the open and
closed subsets, respectively, of a topological space. Another important
special case is that in which
C
is the set of linear subspaces of a vector space
V; in this case, the mapping
span : ℘(V) →
℘(V)
sending each subset of V to its span is a
corresponding closure.

An adjunction between two posets P and Q
is an ordered pair (f, g) of mappings
f : P → Q and
g : Q → P connected by
the condition that, for all p ∈ P, q
∈ Q

f(p)
⊴
q if and
only if p
⊴
g(q).

In this case, we call f a left adjoint for
g, and call g a right adjoint for
f. Two basic facts about adjunctions, both easily proved, are
the following: